We consider the logistic regression with Six City data set [1]}, which was studied in [2]} and [3]} in the context of variational Bayes. The data consist of binary responses \(y_{ij}\)  which is the wheezing status (1 if wheezing, 0 if not wheezing) of the \(i\) th child at time-point \(j\) , where \(i=1,\dots ,537\)  (representing 537 children) and \(j=1,2,3,4\)  (corresponding to \(7,8,9,10\)  year-old). Covariates are \(A_{ij},\)  the age of the \(i\) th child at time-point \(j\) , and \(S_i\)  the \(i\) th maternal smoking status (0 or 1). For the numerical stability, the covariates \(A_{ij}\)  are centered at 9 years, i.e., \(A_{ij}=j-3\)  for \(j=1,\dots ,4\)  and all \(i\) . We consider the logistic regression model with a random intercept: \(y_{ij}\vert \beta ,\alpha \sim \mathrm {Bernoulli}(p_{ij})\) , where
\(p_{ij}=\frac{\exp (\beta _1+\beta _2A_{ij}+\beta _3S_i+\alpha _i)}{1+\exp (\beta _1+\beta _2A_{ij}+\beta _3S_i+\alpha _i)}\) 
